In the present paper, we prove that the toric ideals of certain $s$-block diagonal matching fields have quadratic Grobner bases. Thus, in particular, those are quadratically generated. By using this result, we provide a new family of toric degenerations of Grassmannians.
We consider Markov chain Monte Carlo methods for calculating conditional p values of statistical models for count data arising in Box-Behnken designs. The statistical model we consider is a discrete version of the first-order model in the response su
rface methodology. For our models, the Markov basis, a key notion to construct a connected Markov chain on a given sample space, is characterized as generators of the toric ideals for the centrally symmetric configurations of root system D_n. We show the structure of the Groebner bases for these cases. A numerical example for an imaginary data set is given.
It is known that a Markov basis of the binary graph model of a graph $G$ corresponds to a set of binomial generators of cut ideals $I_{widehat{G}}$ of the suspension $widehat{G}$ of $G$. In this paper, we give another application of cut ideals to sta
tistics. We show that a set of binomial generators of cut ideals is a Markov basis of some regular two-level fractional factorial design. As application, we give a Markov basis of degree 2 for designs defined by at most two relations.
Hara, Takemura and Yoshida discuss toric ideals arising from two way subtable sum problems and shows that these toric ideals are generated by quadratic binomials if and only if the subtables are either diagonal or triangular. In the present paper, we
show that if the subtables are either diagonal or triangular, then their toric ideals possess quadratic Groebner bases.
The correspondence between unmixed bipartite graphs and sublattices of the oolean lattice is discussed. By using this correspondence, we show the existence of squarefree quadratic initial ideals of toric ideals arising from minimal vertex covers of unmixed bipartite graphs.
Let $G$ be a finite graph allowing loops, having no multiple edge and no isolated vertex. We associate $G$ with the edge polytope ${cal P}_G$ and the toric ideal $I_G$. By classifying graphs whose edge polytope is simple, it is proved that the toric
ideals $I_G$ of $G$ possesses a quadratic Grobner basis if the edge polytope ${cal P}_G$ of $G$ is simple. It is also shown that, for a finite graph $G$, the edge polytope is simple but not a simplex if and only if it is smooth but not a simplex. Moreover, the Ehrhart polynomial and the normalized volume of simple edge polytopes are computed.
We consider testing independence in group-wise selections with some restrictions on combinations of choices. We present models for frequency data of selections for which it is easy to perform conditional tests by Markov chain Monte Carlo (MCMC) metho
ds. When the restrictions on the combinations can be described in terms of a Segre-Veronese configuration, an explicit form of a Grobner basis consisting of moves of degree two is readily available for performing a Markov chain. We illustrate our setting with the National Center Test for university entrance examinations in Japan. We also apply our method to testing independence hypotheses involving genotypes at more than one locus or haplotypes of alleles on the same chromosome.
In this paper we introduce a new and large family of configurations whose toric ideals possess quadratic Groebner bases. As an application, a generalization of algebras of Segre-Veronese type will be studied.
